The problem is that I am making this calculation thousands of time per second in a radar simulation program and if I can help it, I would like to not waste system resources on unnecessary conversions. I found the answer by the way.

For anyone else who might be looking for this solution:

The coordinates of the point dividing the line segment P1P2 in the ratio a/b are:

range = $\displaystyle (sqrt[b*b*r_1*r_1+a*a*r_2*r_2+2*a*b*r_1*r_2*cos(theta_2-theta_1)]/[a+b]$

theta = $\displaystyle arctan([b*r_1*sin(theta_1)+a*r_2*sin(theta_2)]/[b*r_1*cos(theta_1)+a*r_2*cos(theta_2)])$

It looks intimidating, but it isn't. The hardest part is making sure you have the correct sign after calculating arctan(). If you are implementing it in Java, you can use:

Code:

tempTheta = Math.atan2(tempRange, tempAz);
if(tempTheta < 0) tempTheta += (2 * Math.PI);

Math.atan2(y, x) computes the phase *theta* by computing an arc tangent of y/x in the range of -*pi* to *pi. *In the case of being negative, just add 2*PI.

For more on Points and Lines in Polar Coordinates, look here: Math Forum: Ask Dr. Math FAQ: Polar Coordinates

Thanks anyway!